On- and off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces (Q2702997)

The authors look at a wide class of symmetric diffusion semigroups with continuous heat kernel on infinite-dimensional state space. It is shown that one can obtain Gaussian type estimates in terms of some relaxed distance function in cases where the corresponding estimate fails to hold for the intrinsic distance. By characterizing the validity of certain off-diagonal estimates of the heat kernel by potential theoretical methods in terms of harmonic sheaves it is proved that a certain on-diagonal growth condition on the heat kernel is equivalent to some off-diagonal estimate if the corresponding Dirichlet form is regular, strictly local and admits a good algebra.NEWLINENEWLINENEWLINEMoreover, invariant strictly local Dirichlet spaces on the infinite-dimensional torus are treated in detail and it is proved that the above-mentioned equivalence between on-diagonal and off-diagonal estimates also holds in this case.